Complex extension of Wigner's theorem

TL;DR

This paper generalizes Wigner's theorem by showing that complex geodesic-preserving transformations on quantum state spaces are generated by non-Hermitian Hamiltonians, extending the class of symmetry transformations in quantum mechanics.

Contribution

It introduces a complex extension of Wigner's theorem, relaxing Hermiticity and characterizing transformations via holomorphic projective mappings.

Findings

Holomorphic projective transformations are generated by non-Hermitian Hamiltonians.

The generalization broadens the scope of symmetry transformations in quantum theory.

Provides a mathematical framework for non-Hermitian quantum dynamics.

Abstract

Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorphically projective (complex geodesic-curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian.